Best Known (38, 38+34, s)-Nets in Base 81
(38, 38+34, 730)-Net over F81 — Constructive and digital
Digital (38, 72, 730)-net over F81, using
- t-expansion [i] based on digital (36, 72, 730)-net over F81, using
- net from sequence [i] based on digital (36, 729)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 36 and N(F) ≥ 730, using
- the Hermitian function field over F81 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 36 and N(F) ≥ 730, using
- net from sequence [i] based on digital (36, 729)-sequence over F81, using
(38, 38+34, 3151)-Net over F81 — Digital
Digital (38, 72, 3151)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8172, 3151, F81, 2, 34) (dual of [(3151, 2), 6230, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8172, 3289, F81, 2, 34) (dual of [(3289, 2), 6506, 35]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8172, 6578, F81, 34) (dual of [6578, 6506, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- linear OA(8167, 6561, F81, 34) (dual of [6561, 6494, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(8155, 6561, F81, 28) (dual of [6561, 6506, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(815, 17, F81, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,81)), using
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- Reed–Solomon code RS(76,81) [i]
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- OOA 2-folding [i] based on linear OA(8172, 6578, F81, 34) (dual of [6578, 6506, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(8172, 3289, F81, 2, 34) (dual of [(3289, 2), 6506, 35]-NRT-code), using
(38, 38+34, large)-Net in Base 81 — Upper bound on s
There is no (38, 72, large)-net in base 81, because
- 32 times m-reduction [i] would yield (38, 40, large)-net in base 81, but